Optimality of the Wald SPRT for Processes with Continuous Time Parameter

The paper deals with the Wald sequential test of two simple hypotheses for processes with continuous time parameter. The observation is a likelihood ratio process, which is a right-continuous local martingale having left-side limits. The Wald sequential test is proved to be optimal in the sense that it minimizes the Kullback-Leibler information under both hypotheses among all tests having no larger error probabilities. The proof is based on the explicit solution of the related optimal stopping problem.